Geosynchronous orbit: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
No edit summary
en>Nyq
m →Geostationary orbit: Capitalized proper noun
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
{{Redirect|Doublestrike||Double Strike (disambiguation)}}
Hello and welcome. My title is Figures Wunder. Years ago we moved to North Dakota. He utilized to be unemployed but now he is a meter reader. To collect badges is what her family members and her enjoy.<br><br>Check out my blog post at home std testing ([http://www.ddaybeauty.com/node/14766 simply click the up coming article])
{{refimprove|date=May 2010}}
{{SpecialChars}}
[[Image:Blackboard bold.svg|right|thumb|250px|An example of blackboard bold letters.]]
'''Blackboard bold''' is a [[typeface]] style that is often used for certain symbols in [[mathematics|mathematical]] texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote [[Set (mathematics)|number sets]]. Blackboard bold symbols are also referred to as '''double struck''', although they cannot actually be produced by double striking on a [[typewriter]].
 
== Origin ==
 
In some texts these symbols are simply shown in [[bold type]]: blackboard bold in fact originated from the attempt to write bold letters on [[chalkboard|blackboard]]s in a way that clearly differentiated them from non-bold letters, and then made its way back in print form as a separate style from ordinary bold,<ref name="sci.math-thread">[http://groups.google.com/group/comp.text.tex/browse_thread/thread/55bd03b808263cee/6300d1a3a394ccb9 Google Groups]</ref> possibly starting with the original 1965 edition of [[Robert Gunning (mathematician)|Gunning]] and Rossi's textbook on complex analysis.<ref>{{cite book | last1=Gunning | first1=Robert C. | authorlink1=Robert Gunning (mathematician) | last2=Rossi | first2=Hugo | year=1965 | title=Analytic functions of several complex variables | publisher=Prentice-Hall }}</ref> 
 
It is sometimes erroneously claimed<ref name="sci.math-thread" />{{Failed verification|date=November 2013}} <!-- The source does not, as far as I saw, actually assert explicitly that Bourbaki did not originate the notation; simply that they were somewhat inconsistent with it individually and did not use it in the books checked by the participants in the discussion.  I am also worried about the veracity of this source. --> that [[Bourbaki]] introduced the blackboard bold notation, but whereas individual members of the Bourbaki group may have popularized double-striking bold characters on the blackboard, their printed books use ordinary bold.<ref>E.g., {{cite book | last=Bourbaki | first=Nicolas | authorlink=Nicolas Bourbaki | title=Théorie des ensembles | publisher=Herman | year=1970 }}</ref>
 
== Rejection ==
 
Some mathematicians, therefore, do not recognize blackboard bold as a separate style from bold: [[Jean-Pierre Serre]], for example, has publicly inveighed against the use of "blackboard bold" anywhere other than on a blackboard,{{Citation needed|date=March 2011}}. He uses double-struck letters when writing bold on the blackboard,<ref>[http://www.dailymotion.com/video/xf88g3_jean-pierre-serre-writing-mathemati_tech "Writing Mathematics Badly" video talk (part 3/3)], starting at 7&prime;08&Prime;</ref> whereas his published works consistently use ordinary bold for the same symbols.<ref>E.g., {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Cohomologie galoisienne | publisher=Springer-Verlag }}</ref> [[Donald Knuth]] also advises against the use of blackboard bold in print.<ref>Krantz, S., ''Handbook of Typography for the Mathematical Sciences'', Chapman & Hall/CRC, Boca Raton, Florida, 2001, p. 35.</ref>{{not in citation|date=January 2014}}
 
The [[Chicago Manual of Style]] in 1993 (14th edition) advises: "blackboard bold should be confined to the classroom" (13.14) whereas in 2003 (15th edition) it states that "open-faced (blackboard) symbols are reserved for familiar systems of numbers" (14.12).
 
== Encoding ==
 
[[TeX]], the standard typesetting system for mathematical texts, does not contain direct support for blackboard bold symbols, but the add-on AMS Fonts package (<code>amsfonts</code>) by the [[American Mathematical Society]] provides this facility; a blackboard bold '''R''' is written as <code>\mathbb{R}</code>.
 
In [[Unicode]], a few of the more common blackboard bold characters ('''C''', '''H''', '''N''', '''P''', '''Q''', '''R''' and '''Z''') are encoded in the [[Mapping of Unicode characters#Basic Multilingual Plane|Basic Multilingual Plane (BMP)]] in the ''Letterlike Symbols (2100–214F)'' area, named DOUBLE-STRUCK CAPITAL C etc. The rest, however, are encoded outside the BMP, from <code>U+1D538</code> to <code>U+1D550</code> (uppercase, excluding those encoded in the BMP), <code>U+1D552</code> to <code>U+1D56B</code> (lowercase) and <code>U+1D7D8</code> to <code>U+1D7E1</code> (digits). Being outside the BMP, these are relatively new and not widely supported.
 
== Usage ==
 
The following table shows all available Unicode blackboard bold characters.
 
The symbols are nearly universal in their interpretation, unlike their normally-typeset counterparts, which are used for many different purposes.
 
The first column shows the letter as typically rendered by the ubiquitous [[LaTeX]] markup system. The second column shows the Unicode codepoint. The third column shows the symbol itself (which will only display correctly if your browser supports Unicode and has access to a suitable font). The fourth column describes known typical (but not universal) usage in mathematical texts.
 
{| summary="Blackboard Bold Characters" class="wikitable"
![[Image:LaTeX logo.svg|48px|\LaTeX]]
!Unicode (Hex)
!Symbol
!Mathematics usage
|-
|<math>\mathbb{A}</math>
|<code>U+1D538</code>
|<big>{{Unicode|𝔸}}</big>
|Represents [[affine space]] or the [[adele ring|ring of adeles]]. Sometimes represents the [[algebraic number]]s, the [[algebraic closure]] of '''Q''' ({{Unicode|ℚ}} or <span style="text-decoration: overline">{{Unicode|ℚ}}</span>, although <span style="text-decoration: overline">'''Q'''</span> is often used instead).  It may also represent the [[algebraic integer]]s, an important subring of the algebraic numbers.
|-
|
|<code>U+1D552</code>
|<big>{{Unicode|𝕒}}</big>
|
|-
|<math>\mathbb{B}</math>
|<code>U+1D539</code>
|<big>{{Unicode|𝔹}}</big>
|Sometimes represents a [[ball (mathematics)|ball]], a [[boolean domain]], or the [[Brauer group]] of a field.
|-
|
|<code>U+1D553</code>
|<big>{{Unicode|𝕓}}</big>
|
|-
|<math>\mathbb{C}</math>
|<code>U+2102</code>
|<big>{{Unicode|ℂ}}</big>
|Represents the [[complex number]]s.
|-
|
|<code>U+1D554</code>
|<big>{{Unicode|𝕔}}</big>
|
|-
|<math>\mathbb{D}</math>
|<code>U+1D53B</code>
|<big>{{Unicode|𝔻}}</big>
|Represents the unit ([[open set|open]]) disk in the [[complex plane]] (for example as a model of the [[Hyperbolic geometry|Hyperbolic plane]]), or the decimal fractions (see [[number]]).
|-
|
|<code>U+1D555</code>
|<big>{{Unicode|𝕕}}</big>
|
|-
|<math>D\!\!\!\!D</math>
|<code>U+2145</code>
|<big>{{Unicode|ⅅ}}</big>
|
|-
|<math>d\!\!\!\!d</math>
|<code>U+2146</code>
|<big>{{Unicode|ⅆ}}</big>
|May represent the [[differential (calculus)|differential]] symbol.
|-
|<math>\mathbb{E}</math>
|<code>U+1D53C</code>
|<big>{{Unicode|𝔼}}</big>
|Represents the [[expected value]] of a [[random variable]], or [[Euclidean space]], or a [[Field (mathematics)|field]] in a [[tower of fields]].
|-
|
|<code>U+1D556</code>
|<big>{{Unicode|𝕖}}</big>
|
|-
|<math>e\!\!e</math>
|<code>U+2147</code>
|<big>{{Unicode|ⅇ}}</big>
|Sometimes used for the mathematical constant ''[[e (mathematical constant)|e]]''.
|-
|<math>\mathbb{F}</math>
|<code>U+1D53D</code>
|<big>{{Unicode|𝔽}}</big>
|Represents a [[Field (mathematics)|field]].  Often used for [[finite field]]s, with a subscript to indicate the order.  Also represents a [[Hirzebruch surface]] or a [[free group]], with a subset to indicate the number of generators (or generating set, if infinite).
|-
|
|<code>U+1D557</code>
|<big>{{Unicode|𝕗}}</big>
|
|-
|<math>\mathbb{G}</math>
|<code>U+1D53E</code>
|<big>{{Unicode|𝔾}}</big>
|Represents a [[Grassmannian]] or a [[group (mathematics)|group]], especially an [[algebraic group]].
|-
|
|<code>U+1D558</code>
|<big>{{Unicode|𝕘}}</big>
|
|-
|<math>\mathbb{H}</math>
|<code>U+210D</code>
|<big>{{Unicode|ℍ}}</big>
|Represents the [[quaternions]] (the H stands for [[William Rowan Hamilton|Hamilton]]), or the [[upper half-plane]], or [[hyperbolic space]], or [[hyperhomology]] of a complex.
|-
|
|<code>U+1D559</code>
|<big>{{Unicode|𝕙}}</big>
|
|-
|<math>\mathbb{I}</math>
|<code>U+1D540</code>
|<big>{{Unicode|𝕀}}</big>
|Occasionally used to denote the [[Identity function|identity mapping]] on an [[algebraic structure]], or the set of [[imaginary number]]s (''i.e.'', the set of all real multiples of the [[imaginary unit]]).
|-
|
|<code>U+1D55A</code>
|<big>{{Unicode|𝕚}}</big>
|
|-
|<math>i\!i</math>
|<code>U+2148</code>
|<big>{{Unicode|ⅈ}}</big>
|Occasionally used for the [[imaginary unit]].
|-
|<math>\mathbb{J}</math>
|<code>U+1D541</code>
|<big>{{Unicode|𝕁}}</big>
|Sometimes represents the [[irrational number]]s, '''R'''\'''Q''' ({{Unicode|ℝ\ℚ}}).
|-
|
|<code>U+1D541</code>
|<big>{{Unicode|𝕛}}</big>
|
|-
|<math>j\!\!j</math>
|<code>U+2149</code>
|<big>{{Unicode|ⅉ}}</big>
|
|-
|<math>\mathbb{K}</math>
|<code>U+1D542</code>
|<big>{{Unicode|𝕂}}</big>
|Represents a [[Field (mathematics)|field]], typically a scalar field. This is derived from the [[German language|German]] word ''Körper'', which is German for field (literally, "body"; cf. the French term ''corps'').  May also be used to denote a [[compact space]].
|-
|
|<code>U+1D55C</code>
|<big>{{Unicode|𝕜}}</big>
|
|-
|<math>\mathbb{L}</math>
|<code>U+1D543</code>
|<big>{{Unicode|𝕃}}</big>
|Represents the [[Lefschetz motive]]. See [[Motive (algebraic geometry)|motives]].
|-
|
|<code>U+1D55D</code>
|<big>{{Unicode|𝕝}}</big>
|
|-
|<math>\mathbb{M}</math>
|<code>U+1D544</code>
|<big>{{Unicode|𝕄}}</big>
|Represents the [[monster group]]. The [[set (mathematics)|set]] of all ''m''-by-''n'' [[Matrix (mathematics)#Notation|matrices]] is denoted 𝕄(''m'', ''n'').
|-
|
|<code>U+1D55E</code>
|<big>{{Unicode|𝕞}}</big>
|
|-
|<math>\mathbb{N}</math>
|<code>U+2115</code>
|<big>{{Unicode|ℕ}}</big>
|Represents the [[natural number]]s. May or may not include [[0 (number)|zero]].
|-
|
|<code>U+1D55F</code>
|<big>{{Unicode|𝕟}}</big>
|
|-
|<math>\mathbb{O}</math>
|<code>U+1D546</code>
|<big>{{Unicode|𝕆}}</big>
|Represents the [[octonion]]s.
|-
|
|<code>U+1D560</code>
|<big>{{Unicode|𝕠}}</big>
|
|-
|<math>\mathbb{P}</math>
|<code>U+2119</code>
|<big>{{Unicode|ℙ}}</big>
|Represents [[projective space]], the [[probability]] of an event, the [[prime number]]s, a [[power set]], the positive reals, the [[Irrational number#The set of all irrationals|irrational numbers]], or a [[forcing (mathematics)|forcing]] [[partially ordered set]] (poset).
|-
|
|<code>U+1D561</code>
|<big>{{Unicode|𝕡}}</big>
|
|-
|<math>\mathbb{Q}</math>
|<code>U+211A</code>
|<big>{{Unicode|ℚ}}</big>
|Represents the [[rational number]]s. (The Q stands for [[quotient]].)
|-
|
|<code>U+1D562</code>
|<big>{{Unicode|𝕢}}</big>
|
|-
|<math>\mathbb{R}</math>
|<code>U+211D</code>
|<big>{{Unicode|ℝ}}</big>
|Represents the [[real number]]s.
|-
|
|<code>U+1D563</code>
|<big>{{Unicode|𝕣}}</big>
|
|-
|<math>\mathbb{S}</math>
|<code>U+1D54A</code>
|<big>{{Unicode|𝕊}}</big>
|Represents the [[sedenion]]s, or a [[sphere]].
|-
|
|<code>U+1D564</code>
|<big>{{Unicode|𝕤}}</big>
|
|-
|<math>\mathbb{T}</math>
|<code>U+1D54B</code>
|<big>{{Unicode|𝕋}}</big>
|Represents a [[torus]], or the [[circle group]], or a [[Hecke algebra]] (Hecke denoted his operators as ''T''<sub>''n''</sub> or {{Unicode|𝕋}}<sub>{{Unicode|𝕟}}</sub>), or the [[Tropical geometry|Tropical semi-ring]], or [[twistor space]].
|-
|
|<code>U+1D565</code>
|<big>{{Unicode|𝕥}}</big>
|
|-
|<math>\mathbb{U}</math>
|<code>U+1D54C</code>
|<big>{{Unicode|𝕌}}</big>
|
|-
|
|<code>U+1D566</code>
|<big>{{Unicode|𝕦}}</big>
|
|-
|<math>\mathbb{V}</math>
|<code>U+1D54D</code>
|<big>{{Unicode|𝕍}}</big>
|Represents a [[vector space]].
|-
|
|<code>U+1D567</code>
|<big>{{Unicode|𝕧}}</big>
|
|-
|<math>\mathbb{W}</math>
|<code>U+1D54E</code>
|<big>{{Unicode|𝕎}}</big>
|Represents the [[Natural number|whole numbers]] (here in the sense of non-negative integers), which also are represented by ''N''<sub>0</sub>.
|-
|
|<code>U+1D568</code>
|<big>{{Unicode|𝕨}}</big>
|
|-
|<math>\mathbb{X}</math>
|<code>U+1D54F</code>
|<big>{{Unicode|𝕏}}</big>
|Occasionally used to denote an arbitrary [[metric space]].
|-
|
|<code>U+1D569</code>
|<big>{{Unicode|𝕩}}</big>
|
|-
|<math>\mathbb{Y}</math>
|<code>U+1D550</code>
|<big>{{Unicode|𝕐}}</big>
|
|-
|
|<code>U+1D56A</code>
|<big>{{Unicode|𝕪}}</big>
|
|-
|<math>\mathbb{Z}</math>
|<code>U+2124</code>
|<big>{{Unicode|ℤ}}</big>
|Represents the [[integer]]s. (The Z is for ''Zahlen'', which is German for "numbers".)
|-
|
|<code>U+1D56B</code>
|<big>{{Unicode|𝕫}}</big>
|
|-
|
|
|
|
|-
|
|<code>U+213E</code>
|<big>{{Unicode|ℾ}}</big>
|
|-
|
|<code>U+213D</code>
|<big>{{Unicode|ℽ}}</big>
|
|-
|
|<code>U+213F</code>
|<big>{{Unicode|ℿ}}</big>
|
|-
|
|<code>U+213C</code>
|<big>{{Unicode|ℼ}}</big>
|
|-
|
|<code>U+2140</code>
|<big>{{Unicode|⅀}}</big>
|
|-
|
|
|
|
|-
|
|<code>U+1D7D8</code>
|<big>{{Unicode|𝟘}}</big>
|
|-
|
|<code>U+1D7D9</code>
|<big>{{Unicode|𝟙}}</big>
|Often represents, in [[set theory]], the [[Greatest element|top element]]  of a forcing [[partially ordered set]] (poset), or occasionally for the  identity matrix in a [[matrix ring]]. Also used for the [[indicator function]].
|-
|
|<code>U+1D7DA</code>
|<big>{{Unicode|𝟚}}</big>
|
|-
|
|<code>U+1D7DB</code>
|<big>{{Unicode|𝟛}}</big>
|
|-
|
|<code>U+1D7DC</code>
|<big>{{Unicode|𝟜}}</big>
|
|-
|
|<code>U+1D7DD</code>
|<big>{{Unicode|𝟝}}</big>
|
|-
|
|<code>U+1D7DE</code>
|<big>{{Unicode|𝟞}}</big>
|
|-
|
|<code>U+1D7DF</code>
|<big>{{Unicode|𝟟}}</big>
|
|-
|
|<code>U+1D7E0</code>
|<big>{{Unicode|𝟠}}</big>
|
|-
|
|<code>U+1D7E1</code>
|<big>{{Unicode|𝟡}}</big>
|
|}
 
In addition, a blackboard-bold Greek letter [[mu (letter)|mu]] (not found in Unicode) is sometimes used by number theorists and algebraic geometers (with a subscript ''n'') to designate the group (or more specifically [[group scheme]]) of ''n''-th [[root of unity|roots of unity]].<ref>{{cite book | last=Milne | first=James S. | title=Étale cohomology | publisher=Princeton University Press | year=1980 | page=xiii }}</ref>
 
==See also==
*[[Mathematical alphanumeric symbols]]
*[[Set notation]]
 
==References==
*{{MathWorld|title=Doublestruck|urlname=Doublestruck}}
<references/>
 
==External links==
*http://www.w3.org/TR/MathML2/double-struck.html shows blackboard bold symbols together with their Unicode encodings. Encodings in the Basic Multilingual Plane are highlighted in yellow.
 
[[Category:Mathematical notation]]
[[Category:Mathematical typefaces]]

Latest revision as of 11:16, 2 January 2015

Hello and welcome. My title is Figures Wunder. Years ago we moved to North Dakota. He utilized to be unemployed but now he is a meter reader. To collect badges is what her family members and her enjoy.

Check out my blog post at home std testing (simply click the up coming article)

Retrieved from "https://en.formulasearchengine.com/index.php?title=Geosynchronous_orbit&oldid=221672"